Conditional expectation and probability

Conditional expectation

D4.1.1 Let $(Ω, ℱ, P)$ be a ps, and $ \subset ℱ$ a sub- $σ$ -field. Let $X$ be a random variable with $E | X | < \infty$ , then the conditional expectation of $X$ given $$ , $E [X | ] : Ω \to ℝ$ and:

$E (X | )$ is $$ -measurable
$\int_{A} E (X | ) d P = \int_{A} X d P \forall A \in $

Remarks:

$Ω \in $ so $E (E (X | )) = E (X)$
any $$ measurable function which differs from $E (X | )$ on a set of measure 0 also qualifies as a conditional expection of $X$

P4.1.1 For any rv $X$ with $E | X | < \infty$ and a sub- $σ$ -field $$ , $E (X | )$ exists and is unique wp1.

P4.1.2 If $X$ an rv $E | X | < \infty$ , $$ a sub $σ$ -field.
If $Z$ is an integrable $$ -measurable rv st for a $π$ -class $D$ with $σ (D) = $ $E X = E Z$ and $\int_{A} Z d P = \int_{A} X d P \forall A \in D$ then $Z = E (X | ) wp1$ .

C4.1.3 Let $_{1}, _{2}$ be sub $σ$ -fields of $ℱ$ and $X, Y$ be integrable rvs.

if $σ (X)$ and $_{1}$ are indepdendent, then $E (X | G_{1}) = E (X) wp1$
if $\int_{A_{1} \cap A_{2}} X d P = \int_{A_{1} \cap A_{2}} Y d P \forall A_{i} \subset _{i}$ , then $E (X | σ (_{1} \cup _{2})) = E (Y | σ (_{1} \cup _{2}))$

P4.1.4 (Properties of CE). Let $X, Y$ be random variables on $(Ω, ℱ, P)$ , $ \subset ℱ$ a sub- $σ$ -field. Then:

$E (1 | ) = 1$
$E (X | ) \geq 0 wp1$ if $X \geq 0 wp1$
$E (c X | ) = c E (X | ) wp1$ if $| E X | \leq \infty$ , $c \in ℝ$
$E (X + Y | ) = E (X | ) + E (Y | ) wp1$ if $E | X + Y | \leq \infty$
$E (X | ) = X wp1$ if $X$ is $$ -measurable
If $_{1} \subset _{2} \subset F$ , $| E X | \leq \infty$ then $E (E (X | G_{2}) | G_{1}) = E (X | G_{1}) = E (E (X | G_{1}) | G_{2})$

T4.1.5 $X$ an rv, $| E X | < \infty$ and $ \subset ℱ$ . If $Y$ is a finite valued $$ -measurable random variable st $| E (X | Y) | \leq \infty$ , then $E (X Y | ) = Y E (X | ) wp1$ .

T4.1.6 ${X_{n}}$ , Y be random variables with $E | Y | < \infty$ and $ \subset ℱ$ on $(Ω, ℱ, P)$ then:

(MCT) If $Y \leq X_{n} ↑ wp1$ then $E (X_{n} | ) \to E (X | ) wp1$
(Fatou) If $Y \leq X_{n} \forall n \geq 1 wp1$ then $E (lim_{̲} X_{n} | ) \leq lim_{̲} E (X_{n} | ) wp1$
(DCT) If $X_{n} \to^{wp1} X$ and $| X_{n} | \leq | Y | wp1 \forall n \geq 1$ then $E (X_{n} | ) \to E (X | ) wp1$

T4.1.7 (Jensen). $Y$ a $ℱ$ -measurable with $| E Y | \leq \infty$ and $g$ be a finite convex function on $ℝ$ with $| E g (Y) | \leq \infty$ . If for some $σ$ -field $$ :

$X = E (Y | ) wp1$ OR
$X$ is $ℱ$ -measurable with $X \leq E (X | ) wp1$ and $g ↑$

then $g (X) \leq E (g (X) |  wp1$

D4.1.2 If $E Y^{2} < \infty$ , the conditional variance of $Y$ given sub- $σ$ -field $$ $var (Y | ) = E (Y^{2} | ) + E^{2} (Y | )$

T4.1.8 Let $E Y^{2} < \infty$ then $var (Y) = var (E (Y | )) + E (var (Y | ))$

Conditional probability

D4.2.1 Let $(Ω, ℱ, P)$ be a ps. For a $B \in ℱ$ and sub $σ$ -field $ \subset ℱ$ , the conditional probability of $B$ given $$ , $P (B | ) = E (I_{B} | )$ .

Remarks:

conditional probability is a just a conditioanl expection of an indicator rv, satisfying:
- $P (B | )$ is $$ -measurable
- $\int_{A} P (B | ) d P = \int_{A} I_{B} d P = \int I_{A \cap B} d P \forall A \in $ , ie. $P (A \cap B) = E (P (B | ) I_{A})$

The properties of conditional expectation lead to the following properties of conditional probability:

$0 \leq P (A | ) \leq 1 wp1$
$P (A | ) = 0 wp1 \Leftrightarrow P (A) = 0$
$P (A | ) = 1 wp1 \Leftrightarrow P (A) = 1$
If ${A_{n}} \subset ℱ$ are disjoint sets, then $P (U U_{i} A_{i} | ) = \sum_{i} P (A_{i} | ) wp1$
If $A_{n} \in F n \geq 1$ and $lim_{n \to \infty} A_{n} = A$ then $lim_{n \to \infty} P (A_{n} | ) = P (A | ) wp1$

Above suggests that $μ_{} (\cdot) = P (\cdot | )$ is sub-additive wp1 for a given set of ${A_{i}}$ . However, this may not be subadditive in general wp1, as the probability 1 set may change from set to set. Hence we may not be able to find a common probability 1 set $\Rightarrow P (\cdot | )$ may not be a pm on $ℱ$ .

D4.2.2 Let $ℱ_{1}, $ be sub- $σ$ -fields of events in $(Ω, ℱ, P)$ . A regular conidtional probability on $F_{1} | $ is a function $μ : ℱ_{1} \times Ω \to [0, 1]$ satisfying:

$\forall B \in ℱ_{1}$ , $μ (B, ω) = P (B | ) (ω)$
$\forall ω \in Ω$ , $μ (\cdot, ω)$ is a pm on $ℱ_{1}$

T4.2.1 Let $P_{ω} (A) := P (A, w)$ be a regular conditional probability. Given $$ , $X$ is $ℱ$ -measuralbe with $| E X | \leq \infty$ then $E (X | ) (ω) = \int_{Ω} X d P_{ω}$